#8581788

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,7,8,5)(10,13)(11,12)(17,21)(18,20)(25,26,32,35)(27,36,38,33)(28,31,29,39)(30,34,40,37);(1,3,4,8)(2,5,6,7)(9,10,15,13)(11,16,12,19)(14,17,22,21)(18,23,20,24)(25,27)(26,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,16,13,19)(14,18,22,20)(17,23,21,24)(25,28,32,29)(26,31,35,39)(27,34,38,37)(30,36,40,33);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,13,19)(14,18,22,20)(17,24,21,23)(25,29,32,28)(26,31,35,39)(27,37,38,34)(30,36,40,33);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,16,12,19)(14,21,22,17)(18,23,20,24)(25,30)(26,34)(27,31)(28,33)(29,36)(32,40)(35,37)(38,39);(9,14,15,22)(10,17,13,21)(11,18,12,20)(16,23,19,24)(25,31)(26,29)(27,30)(28,35)(32,39)(33,37)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488984);